1
MEASURING SYSTEMATIC RISK USING
ASYMMETRIC AND FAT-TAILED
RETURN DISTRIBUTIONS
WITH AN APPLICATION IN
MINIMUM VARIANCE PORTFOLIO CONSTRUCTION
Hao Li
University of Groningen Faculty of Economics and Business
Econometrics&Finance Supervisor Prof. Dr. Theo.K.Dijkstra
July 2014
ABSTRACT
In this paper, normal return distributions are pitted against two families of alternative distributions, which are both capable of accommodating asymmetry and fat tails: the Asymmetric Power Distributions and the Asymmetric Exponential Power Distributions. The Capital Asset Pricing Models for three different return distributions are fitted using maximum likelihood of daily, weekly, and monthly returns of the EURO STOXX 50 index and its constituents from the period of 2/1/2009 to 3/1/2014. This analysis is believed to be a novel approach. The power distributions fit the daily and weekly data best, as measured by the Akaike Information Criterion (AIC) and as judged by density plots, whereas the normal distribution appears to be best for the monthly data.
The models also yield estimates for beta, which is the measure of systematic risk in comparison to the market as a whole. These estimates are essential input for the estimated covariance matrix of the returns for the single-index model (SIM). Minimum Variance Portfolios with non-negative weights are determined recursively for monthly data, using all three return distributions, re-calculating all estimates and weights anew for every month that is added to the data set. This yields five years of honest, monthly returns for each alternative. For the monthly frequency, the results for all three distributions are quite comparable.
JEL classifications: C58 G12 G17
Key words: Capital Asset Pricing Model; Non-normal distribution; Systematic Risk; Minimum Variance Portfolio
2 1. Introduction
Markowitz (1952; 1959) first proposes the investor's portfolio selection problem regarding to the expected return and the variance of return. The studies of Sharpe (1964), Lintner (1965), Mossin (1966) and Black (1972), which are on the capital asset pricing model (CAPM), extend his work into economy-wide implications. According to Plantinga (2013a), these studies show that if investors have homogeneous expectations and optimally hold mean-variance efficient portfolios then the portfolio of all invested wealth, or the market portfolio, will itself be a mean-variance efficient portfolio in the absence of market frictions. In light of Campbell, Lo and MacKinlay (1997), the usual CAPM equation is a direct implication of the mean-variance efficiency of the market portfolio. Under this theory framework, the normal distribution is widely employed.
However, vast empirical evidences show that returns are not usually normally distributed (see, e.g., Fama, 1965a; Mandelbrot, 1967; Blattberg and Gonedes, 1974; Affleck-Graves and McDonald, 1989). Questioning the normality assumption of asset returns is hardly new. Mandelbrot (1963a; 1963b) investigates the normality assumption in depth and contends that random return processes do follow the stable Paretian distribution. Fama (1963; 1965a; 1965b) also has this interest in studying the empirical distribution of asset returns and establishes a CAPM for symmetric stable Paretian returns (see Fama, 1971). Moreover, Rachev and Mittnik (2000) and Rachev (2004) comprehensively discuss the asset pricing and portfolio theory under the stable Paretian distributions.
Campbell, Lo and MacKinlay (1997) state that accounting for asymmetry and fat-tails of financial data is relevant for asset pricing. Nevertheless, according to Thomas and Gup (2010), the stable Paretian is not the only distribution to model the fat tails. Cootner (1964) argues that the evidence of Paretian distribution security return is too casual. Moreover, not only are the probability density functions (PDFs) of the stable Paretian distributions not analytically expressible1, but the finite higher moments (or at least a finite second moment) of the stable Paretian distributions are also undefined. Campbell, Lo, and MacKinlay (1997) deem that stable Paretian distributions are too fat-tailed2 and that the distribution of return may be a fat-tailed unconditional distribution consisting of both finite variance and finite higher moments. Moreover, Thomas and Gup (2010) argue that stock return empirical behavior is inconsistent with the stable Paretian theory of constant “alpha peakedness” and “beta skewness” over different time periods.
Hence, the author uses a novel approach to accounting for asymmetry and fat-tails of asset returns to extend the CAPM. Specifically, the author introduces two power distributions, the Asymmetric Power Distribution (APD) and the Asymmetric Exponential Power Distribution (AEPD), to the CAPM, which have relatively desirable mathematical property of stability. Perhaps the first published
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According to Campbell, Lo, and MacKinlay (1997), the closed-form expressions for density functions of stable Paretian random variables are available only for three special cases: the normal, the Cauchy, and the Bernoulli cases.
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study to extend the CAPM with the effect of skewness on asset pricing is Kraus and Litzenberger (1976), which is a kind of three-fund separation model. Zeckhauser and Thompson (1970) conclude that estimating the power distribution parameters generating the errors is desirable and that its effect on the estimates of regression coefficients cannot be small. Jin (2011) 3 proposes an asset-pricing model called the AEPD, which nests the AEPD in the errors, and concludes that the CAPM-AEPD is better than the CAPM. However, under the framework of the CAPM, the mean of residuals should be zero. As a result, two new asset-pricing models for measuring the systematic risk are developed in this paper: the Asymmetric Power CAPM (APD-CAPM) and the Asymmetric Power Exponential CAPM (AEPD-CAPM).
The APD and the AEPD are employed because they are more flexible and can yield more distributions. For instance, the AEPD can be the Asymmetric Laplace Distribution (ALD), the Skewed Exponential Power Distribution (SEPD) and the Exponential Power Distribution (EPD), etc. The APD contains the Generalized Power Distribution (GPD)4, the ALD, and the two-piece normal distribution, etc. Table 1 shows the distributions included in the APD and the AEPD. However, Zhu and Zinde-Walsh (2009) argue that the APD is a sort of SEPD and the skewness parameter in the SEPD cannot be rich enough to capture all of the asymmetry of distributions of asset returns, especially the asymmetry in the tails, because the left tail is always thinner than the right one in the SEPD. Hence, other than the APD, the AEPD is employed to investigate which distribution can better capture features of financial data, like asymmetry, heavy tails, and leptokurtic or fat-tails.
Table 1
Special cases of the APD and the AEPD
This table presents the non-normal distributions nested in the APD and the AEPD; The GPD allows flexibility in modeling the tail behavior. According to Komunjer (2007), the GPD can be the uniform distribution when , be short-tailed distributions when , and be the fat-tailed ones when . According to Zhu and Zinde-Walsh (2009), the APD is a sort of SEPD due to the quantification of asymmetry; however, the AEPD can be the SEPD if = .
Distributions Parameters in APD Parameters in AEPD Normal GPD , Laplace Asymmetric Laplace Two-piece normal SEPD -
Concerning the parameters that describe the asymmetric and fat-tailed distributions, measures the skewness and , and are the tail parameters. According to Zhu and Zinde-Walsh (2009), the
3 In Jin (2011), the form of CAPM-AEPD is ( ) , which does not imply that the average of residuals is zero. Moreover, Jin (2011) does not denote the meaning of and . In fact, and are respectively the location and the scale parameter according to his likelihood function. The reference is Jin, H., 2011.Analysis of CAPM based on asymmetric exponential power distribution. Master dissertation. Nankai University. 4
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SEPD and the AEPD skew to the right if and to the left if , controls the left tail and controls the right tail. Specifically, leads to a fat left tail, fatter and fatter for smaller and smaller . Fig.1 shows the density plots of AIR LIQUIDE. The AEPD can better capture the asymmetry in the tails ( ̂ , and ̂ in Table 3 while ̂ in Table 2, see Section 4.2).
Fig.1 The PDF plots of residuals and the given distribution for AIR LIQUIDE
The black solid line represents the empirical distribution while the red dotted line represents the given distribution. The sample period is from 2/1/2009 to 3/1/2014. Take AIR LIQUIDE for example, compared with the upper density plot, the APD and the AEPD can better capture the leptokurtic or fat-tails feature and the AEPD fits the empirical distribution best concerning the tail shape. This can be supported by the AIC values (see Penal A of Table F in Appendix F).
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For the Eurozone stock market, the models discussed in this paper analyze the total returns of the EURO STOXX 50 index (as the proxy of the market portfolio) and of its constituents. Regarding to the estimation results and the statistical evidence, the data sample from 2/1/2009 to 3/1/2014 includes the daily, weekly, and the monthly returns. The results show that the APD-CAPM and the AEPD– CAPM proposed in this paper have better in-sample fit for the Eurozone stock market than the CAPM, especially for the daily and the weekly returns. Specifically, the APD-CAPM is the best model concerning the daily returns due to 35 out of 50 smallest AIC values and concerning the weekly returns due to 28 out of 50 smallest AIC values (the AEPD–CAPM with only 15 and with only 20, respectively). However, regarding to the monthly return the CAPM performs best, since 29 AIC values are the smallest. Moreover, for the daily and weekly returns, most of the estimates that describe the APD and the AEPD are significant. Besides the normality test, this paper also performs the Laplace distribution test judged by the 95% confidence intervals of the skewness parameters and the tail parameters and more results from the AEPD-CAPM are significant than those from the APD-CAPM.
In terms of the application of the systematic risk, the minimum variance portfolios (MVPs) are constructed because they can perform well in practice and avoid the risk in estimating the expected returns. The recursive analysis is used to see the performance of the MVPs constructed by using the single-index model (for the covariance matrix estimate).The different distribution types of CAPMs analyze the constituents of the EURO STOXX 50 index only except for GDF SUEZ due to data availability. The data sample is from 14/7/2001 to 14/4/2014. The minimum variance portfolio results based on the ex ante beta estimates show that some periods’ Sharpe ratios via the asymmetric CAPMs5
are higher (26 via the APD-CAPM approach and 19 via the AEPD-CAPM approach) than those via the CAPM, although the CAPM is best model for monthly returns (see the results in Section 4.3.1). Moreover, there are 29 out of 59 periods’ minimum variance portfolio returns generated by the asymmetric CAPMs higher than those generated by the CAPM. Furthermore, employing the beta estimates of the APD-CAPM and the AEPD-CAPM can reduce the volatility of the constructed MVPs’ monthly returns. In addition, the sums of MVPs log returns (including dividends) constructed by the asymmetric CAPMs are higher than the one constructed by the CAPM.
Based on these comparable results, there are two open research questions on this subject: (1) what effect does the daily, weekly, and monthly frequency have on the minimum variance portfolio returns and (2) in what way can the bootstrap be used to assess the statistical reliability of the recursive analysis.
This thesis is organized as follows: the theory and methodology are presented in Section 2. Section 3 gives the mathematics details of the CAPMs based on the method of ML. Section 4 shows the results and the statistical evidence, specifically, the proxy choice, the data description, the estimation results
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and the goodness of fit of models (judged by the AIC values and the density plots using the Kernel Density Estimation). Section 5 shows the results of MVPs calculated by the real returns using the recursion programming. Section 6 is the conclusion and discussion.
2. Theory and methodology
According to Campbell, Lo and MacKinlay (1997), the original formulation of the CAPM employs the assumption that returns are IID normality. This is mainly because the normal distribution is statistically convenient to apply with only two describing parameters. However, the returns on some assets are usually not normally distributed(as noted in Section 1). Therefore, the author employs the APD and the AEPD in the asset-pricing model-building process, using the method of ML. Section 2.1 offers an overview of CAPM. Section 2.2 shows the method of ML and its computer implementation. Section 2.3 briefly introduces the APD and the AEPD.
2.1. CAPM
By assuming the existence of lending and borrowing at a risk-free rate of interest, the CAPM is defined as follows:
[ ] [ ] (1) where [ ] is the expected return of asset , [ ] is the expected return on the market portfolio, and is the return on the risk-free asset. [ [ ]]. In equilibrium, the CAPM can quantify risk
and its reward for bearing risk and thus any risky asset can be priced according to its relevant measure of risk, i.e. the systematic risk, .
Eq. (1) can be compactly expressed in terms of excess returns (i.e. returns in excess of the risk-free rate) as follows:
[ ] [ ] (2)
where represents the return on the ith asset in excess of the risk-free rate, . represents the excess return on the market portfolio of assets, . [ [ ]].
According to Campbell, Lo and MacKinlay (1997),when the risk-free rate is treated as being non-stochastic, in Eq.(2) equals in Eq.(1) , however, in empirical implementations, proxies for the risk-free rate are stochastic and thus the betas in Eq.(1) and in Eq.(2) differ. Campbell, Lo and MacKinlay (1997) state that most empirical works relating to the CAPM employ excess returns. Therefore, the author occupies the form of the CAPM defined in Eq. (2) and thus theoretically equals [ [ ]].
Under the statistical framework for estimation, the data-generating process (DGP) of the CAPM6 is as follows:
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(3)
where i denotes the individual asset and t denotes the time period, t = 1,…, T. are the realized excess returns in time period t for the asset i and the market portfolio, respectively.
, , represents the asset excess return intercept and is the disturbance
term. is the proportionality factor that reflects the sensitivity of asset relative to the market risk. In light of Elton et al (2011), the DGP of the different distribution types of CAPMs is constructed by:
1. The residuals are assumed to be independent and identically distributed, and the mean of , [ ] ;
2. [ ] , ;
3. [ ]
As mentioned by Elton et al (2011), the time series regression analysis is one technique to guarantee that and are uncorrelated over the period to which the equation has been fit. Thus, all models in this paper based on the method of ML take the form of Eq. (3). Assumption 1 is in line with the theory of the CAPM. Assumption 2 implies that stocks’ excess returns systematically move together is only caused by a common co-movement with the market. Assumption 3 means the excess return in time period t for the market portfolio, , is uncorrelated to the unique return , which implies that how well Eq.(3) describes the excess return on any asset is independent of what the excess return on the market happens to be in period.
Then the definitions are as follows: 1. the variance of , [ ]= ; 2. [ ] , [ ] ;
3. [ ]
Hence, the results7 of are as follows: 1. The mean of ; 2. The variance of , = ;
3. The covariance between and ,
These results show that the mean or the variance of includes the unique part ( or ) and a
market related part ( or ). However, the covariance relies only on the market. This is because the assumption made earlier, [ ] which is the constraint in estimation. Since the full
covariance matrix estimated without restrictions is unstable from the statistical perspective.
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2.2. The method of Maximum Likelihood and its computer implementation
There is an increasing agreement that the return distributions on some securities and portfolios are not normal (see Sharpe, 2007a), even though most techniques in econometrics assume that they are (see Campbell, Lo, and MacKinlay ,1997). In addition, Brooks (2008) argues that linear structural models, such as Eq.(3) in this paper, are unable to explain a number of important features common to much financial data, like leptokurtic. Moreover, as Campbell, Lo, and MacKinlay (1997) state, the distribution of return may be a fat-tailed unconditional distribution with finite variance and finite higher moments.
Hence, considering the features of return distributions, different distribution types of CAPMs discussed in this paper employ the method of Maximum Likelihood (ML). The reasons are that the ML estimation is a way to find parameter values for both linear and non-linear models and that distributional assumptions of random variables are flexible under the ML framework, which only depend on their probability density functions. Moreover, the ML estimators are asymptotically consistent, efficient, unbiased and normally distributed, which are referred to the best asymptotic normal (BAN) estimators.
Recall that one assumption of the CAPM’s DGP is the mean of = [ ] (as noted in Section 2.1). Therefore, the residual, , in Eq.(3) for a single period can be written as:
(4)
where is the standard deviation of the residual and denotes a random variable,
denotes its mean and denotes its standard deviation. Note that the random variable is standard normal in the traditional CAPM. Similarly, is a standard APD random variable in the APD-CAPM while is a standard AEPD random variable in the AEPD-CAPM. The PDF of the random variable and its mean and variance are given in Section 2.3.
For a random return distribution sample, with , which is IID distributed as
, the log-likelihood to be maximized with respect to the parameters equals
∑ { ( (
))}
where is the PDF of the random variable .
The different distribution types of CAPMs discussed in this paper can be easily implemented through the computer. The optimization toolbox8 in MATLAB R2013a is used to maximize the likelihood. The numerical optimization algorithm is the BFGS algorithm. This algorithm is occupied because the approximation of the Hessian updated by the BFGS algorithm is particularly robust (see
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Hurn, 2009)and because the BFGS algorithm is preferred over successive iterations (see Brooks, 2008).
2.3. Distribution choices
As the benchmark, the author chooses the normal distribution firstly (i.e. the CAPM is initially considered), and then employs the APD and the AEPD to extend the CAPM. However, the normal distribution is a very common distribution. Furthermore, using the method of ML to estimate the normally distributed CAPM is already extensively discussed in Campbell, Lo and MacKinlay (1997), like the joint PDF of , the ML estimators of parameters, and the statistic of the Wald test,
finite-sample F test, likelihood ratio test, and etc. Moreover, the finite finite-sample issues using a Wald-type test are developed by MacKinlay (1987) and Gibbons, Ross, and Shanken (1989). Thus, this paper only introduces the APD and the AEPD because these two distributions are relatively novel.
2.3.1. The Asymmetric Power Distribution
Komunjer (2007) constructs a new family of densities, the APD, to extend the GPD in the sense of capturing the asymmetry of data. This is because the APD family of distributions uses two parameters, and , to combine the asymmetry with the tail decay property of the GPD. Since special cases of the APD include the normal distribution when parameters values of and are given by . Hence, the author establishes a CAPM for APD returns, which is called the Asymmetric Power CAPM, and thus random variable in Eq.(6) has the density of the standard APD.
According to Komunjer (2007), the probability density function ， is defined as follows: { ( ⁄ ⁄ ) [ | | ] ( ⁄ ⁄ ) [ | | ]
(6)
where , , , and . The parameter measures the degree
of asymmetry. The parameter controls the tail decay. is the gamma function.
Komunjer (2007) states that and ∫ , so that is a probability density and continuously differentiable on . Hence, any random variable with density is the standard APD random variable.
According to Komunjer (2007), the mean and the variance of are as follows:
[ ]
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where is the skewness parameter, , is the tail parameter, , , and is the gamma function.
In terms of the ML Estimation, Komunjer (2007) proves the property of ML estimators employing the results in Newey and McFadden (1994). Specifically, the maximum likelihood estimate (MLE) in parameter vector has the properties of consistency and asymptotically normality. Hence, the issue of “standard differentiability” assumption or “stochastic differentiability” condition could be left outside its scope, because the author only focuses on the application of the standard APD on CAPM in this paper.
2.3.2. The Asymmetric Power Exponential Distribution
Zhu and Zinde-Walsh (2009) argue that the left tail is always thinner than the right one in the SEPD due to only one tail parameter. Therefore, Zhu and Zinde-Walsh (2009) propose a new distribution family, the AEPD, to generalize the SEPD in the sense of capturing the asymmetry in the tails of financial data. This distribution family has two tail parameters, and , to allow a more flexible tail shape. Thus, the author proposes an asset-pricing model called the Asymmetric Exponential Power CAPM for AEPD returns.
Based on Zhu and Zinde-Walsh (2009), the PDF of a standard AEPD random variable can be defined as follows:
{ ( ) ( | | ) (
) ( | | )
(9)
where is the skewness parameter, , is the left tail parameter, and is the right tail parameter, .The asymmetry can be measured by , and . According to Zhu and Zinde-Walsh (2009), [
] to ensure continuity and if = , [ ⁄ ⁄ ⁄ ] where is the Gamma function. The mean and the variance of
a standard AEPD random variable are as follows: [ ] (10) { [ ]} (11)
where is the skewness parameter, , is the left tail parameter, and is the right tail parameter, . is the Gamma function.
( ) ( ) .
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based on the true values of parameters and are both larger than one, they argue that the restriction is not an impediment in most applications.
3. Models and the Maximum Likelihood Estimation
This section discusses the different distribution types of CAPMs under the ML framework and gives the corresponding likelihood functions.
3.1. CAPM and the Maximum Likelihood Estimation
Using ML for the CAPM is hardly new. Campbell, Lo and MacKinlay (1997) comprehensively and profoundly discuss the ML Estimation and corresponding ML tests for the CAPM. Recall the DGP of CAPM given by Eq. (3), hence the DGP of the CAPM for the Maximum Likelihood Estimation takes the form:
, (12) ~ ( )
,
where denotes the asset , given observations,
are the excess returns on asset and
on the market portfolio, respectively. represents the return intercept , measures the systematic risk of asset , and is the residual, as discussed in Section 2.2,
, where is a random variable and it follows the standard normal distribution and in this case9,
the mean of , i.e. equals 0 and the standard deviation of , i.e. , equals 1. denotes the
standard deviation of . Note in Eq.(12) the mean of the residuals, [ ] , denotes the variance of , and [ ] .
Recall the definitions and results discussed in Section 2.1, the mean of is defined by , and
the mean of is defined by
,
= .The covariance matrix between is asfollows:
[
] (13)
The excess return on the ith asset is IID normally distributed random variable with the mean
and the variance , Since is a random variable and it follows the standard normal distribution, the mean of ,i.e. equals 0 and the standard deviation of ,i.e. ,equals 1, hence Eq. (5), the log- likelihood function, , for ML estimation takes the form:
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where log is the natural logarithm and the parameter vector and To maximize the log–likelihood with respect to the parameters in , ̂ is the solution to the problem where is a parameter set, .
3.2. APD-CAPM and the Maximum Likelihood Estimation
To avoid conceptualized confusion and to keep the consistency of mathematical notation, the Asymmetric Power CAPM is as follows:
, (15) )
,
where denotes the asset , given observations,
are the excess returns on asset and
on the market portfolio, respectively. represents the intercept , measures the systematic risk of asset , and is the residual, can be expressed by the standard APD random variable with the density defined by Eq.(6), denotes the mean of obtained from Eq.(7) ,and denotes the standard deviation of obtained from Eq.(8). denotes the standard deviation of .
Hence, the log-likelihood function, used for ML estimation is as follows:
( [ ⁄ ⁄ ]) (16) ∑ [ | ( ) | ( ) | ( ) | ( )] where , ,
, and , the parameter measures the degree of
asymmetry and the parameter controls the tail decay, which are used to describe the APD return in the Asymmetric Power CAPM. is the gamma function.
Since
, , ,, and , is the parameter vector, ̂ will be the solution to the problem
where is a parameter set, .
3.3. AEPD-CAPM and the Maximum Likelihood Estimation
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denotes the mean of obtained from Eq.(10), and denotes the standard deviation of obtained from Eq.(11). denotes the standard deviation of . For the AEPD returns described by parameters ,
, and , the Asymmetric Exponential Power CAPM is constructed as follows:
, (17) ,
,
where , and , and represents the intercept , measures the systematic risk of asset .The mean of the residuals, [ ] equals 0. Therefore the log-likelihood function is given as follows:
( ) (18) ∑ [ | | ( ) | | ( ) ] where , ( ) ( ) , [ ] , [ ⁄ ⁄ ]
⁄ and is the Gamma function. denotes the parameter vector , and , ,
.Therefore, ̂will be the solution to the problem where is a parameter set, .
4. Results and statistical evidence
The theory of ML estimation provides standard errors, statistical tests, and other useful results for statistical inference. The parameters that best fit data can be found and thus the goodness of fit can be assessed, through the estimation procedure. This section shows the estimation results. The goodness of fit is measure by the AIC values and judged by the density plots.
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considering the 2007-08 financial crisis and the following business cycle, the different distribution types of CAPMs are fitted to daily, weekly and monthly returns of the EURO STOXX 50 index and its constituents over five years from January 2009. In line with the CAPM theory, for assets that make periodic dividend payments, the author uses total return index from Datastream10 to account for capital appreciation and income.
4.1. Proxy choice and data description
In this section, the author discusses the proxies of the market portfolio and the risk-free rate. Data source is Datastream. Stock returns are analyzed since stocks commonly exist in investors’ asset mix as a main asset class.
4.1.1. The proxy of the market portfolio
Campbell, Lo, and MacKinlay(1997) mentioned that a broad-based stock index is usually used as the market portfolio and perhaps the Center for Research in Security Prices (CRSP) market index (equal-weighted or value-weighted) is the most widely used proxy of the market portfolio among academics. The reason is that it is unrealistic to construct the real market portfolio that consists of a weighted sum of every asset category including human capital in the market. However, due to data accessibility, the Standard & Poor’s 500 Index (S&P 500) is typically more applicable to the United States market in practice. For the European market, the EURO STOXX 50 Index is chosen as the proxy of the market portfolio. Because the EURO STOXX 50 Index is highly correlated11 with the S&P 500 and these two markets have the similar market efficiency type. In addition, EURO STOXX 50 Index is viewed as Europe's leading Blue-chip index for the Eurozone if considering the currency consistency. Furthermore, the constituents of this index are supersector leaders in the Eurozone with representativeness and diversification12 , and the interval effect influences big firms less.
4.1.2. The proxy of the risk-free rate
The risk-free rate is the other important issue for modeling expected returns. It should not have default risk and reinvestment risk. The 1 month US T-Bill is typically employed as the risk-free asset in practice. However, there are other choices for the European investors, since a foreign holder also requires compensation for potential foreign exchange risk if using the 1 month US T-Bill. Therefore, the practitioners conventionally use the return on long-term Germany Government Bonds as a good proxy of the risk-free rate for the European market. However, the German government cannot print euros and thus the bonds issued by the German government are not fairly non-risk. In addition, European institutional investors widely perceive the London Interbank Offered Rate (LIBOR) as the proxy of the risk-free rate because it is observable. However, considering the issues of tax, regulatory,
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The author thanks Prof. Auke Plantinga for suggesting this data source. 11
Labuszewski (2013) states that the correlation between the S&P 500 and the Euro STOXX 50 pegs at 0.865.www.cmegroup.com/trading/equity-index/files/Stock_Index_Spreading_0410.pdf. Last access date: 25-05-2014. 12
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and liquidity, the author uses the Euro Interbank Offered Rate (Euribor) as the proxy of the risk-free rate. Besides, the overnight indexed swap (OIS) in Euro is used as the risk-free rate to match daily return. According to Hull (2011), after the global financial crisis 07-08, institutional investors gradually perceive the OIS13 as the risk-free rate for high frequency financial data after this crisis.
4.1.3. Data description
As to the rate of return, it is conventional to use continuous compounded return, i.e. the log return (the natural logarithm) in quant finance. The benefits of using log return include time-additivity, mathematical ease, numerical stability, etc. Thus, return on any asset at time is defined as
(19) where is the total return index value at time and is the total return index value at time , denotes the natural logarithm. The excess return of any asset ,
(20)
where is as defined above and denotes the risk-free rate expressed as annual percentage at
time .
Since the risk-free rate is annualized. For different time intervals over five years, the 1-year OIS rate (in Euro) is divided by the number of actual trading days(161, generally, only except each one in 2011 which is divided by 160) to convert to the daily risk-free rate. Similarly, the annualized 1-week Euribor is divided by 52(except each one in 2010 which is divided by 53 due to 53 weekly returns) to convert to the weekly risk-free rate. The annualized 1-month Euribor is divided by 12 to convert to the monthly risk-free rate.
Table B in Appendix B shows the summary of descriptive statistics of the sample data (exported by EViews 7.2). See Panel A, the daily returns on the EURO STOXX 50 index and its 50 constituents do not follow the Gaussian distribution because the p-value of Jarque-Bera statistic for each return series equals zero. Panel B shows that not every weekly return series follows the normal distribution, because there are 20 return series’ p-values of Jarque-Bera statistic strictly equal to zero with the non-zero skewness values and their kurtosis values are larger than 3. However, the normality hypotheses of 7 return series, Bayer (XET), Deutsche Telekom (XET), Essilor Intl, Orange, Philips Eltn.Koninklijke, Telefonica, and Total, cannot be rejected at the 5% significance level. In terms of the monthly returns summarized in Panel C, the total returns on 9 stocks, AXA, Deutsche Bank (XET), Deutsche Post(XET), ENI, ING Groep, Muenchener Ruck (XET), Philips Eltn.Koninklijke, Siemens (XET),and UniCredit, are not normally distributed, since the p-values of Jarque-Bera statistic strictly equal zero. However, 20 stocks’ monthly total returns are normally distributed at the 5% significance level.
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4.2. The estimates and their confidenceintervals
According to Davidson and MacKinnon (2004), minus the inverse of the Hessian is one way to estimate the covariance matrix of the ML estimator, evaluated at the vector of ML estimates due to the consistency of the estimates. Recall that the estimates of parameters are consistent (as noted in Sections 2.3.1 and 2.3.2), we therefore can evaluate the Hessian at ̂.It yields the empirical Hessian estimator, which can be easily obtained by the quasi-Newton methods for maximization. Since the BFGS algorithm (as noted in Section 2.2) is a sort of quasi-Newton method, the Hessian yielded by this algorithm should converge to the “correct” Hessian 14. Hence, the estimate ford standard deviation, i.e. the standard error (SE) evaluated at ̂ is calculated as
SE ̂ =√ ̂ (21) where ̂ denotes the solution to the problem , is the parameter set, SE ̂ denotes the estimated standard deviation vector of ̂, ( ̂) denotes theHessian at ̂.
Concerning the confidence intervals of estimates, the normal distribution is employed to calculate the confidence intervals. The reason is that the estimates of both the coefficients of regression models and the parameters of the distributions are asymptotic normally distributed under the framework of ML estimation. For deciding the statistical significance, the empirical 95% confidence interval (CI) of any estimate is constructed as follows:
CI= ̂ ̂ ̂ ̂ (22) where the solution ̂ is the parameter vector, SE ̂ denotes the estimated standard deviation vector of ̂.
The CI of the estimate also plays an important role regarding to distribution test. Recall in Section 2.3.1, the special cases of the APD include the normal distribution when equals and equals . Hence, if 0.5 lies outside the CI of ̂ or 2 lies outside the CI of ̂ , say statistically significant and thus reject the null hypothesis of the normal distribution at the 5% significance level. Similarly, for the results of AEPD-CAPM, if 2 lies outside the CI of
̂
or̂
, or 0.5 lies outside the CI of ̂ , say significantly not normally distributed with a 95% confidence level. Besides, some non-normal distributions nested in the APD and the AEPD can be tested by the CI and thus the Laplace distribution test is performed later in this section.In addition, the basic empirical test of the CAPM is to estimate Eq. (12) in Section 3.1 and then test if the intercept is zero, i.e. . According to Söderlind (2013), the economic importance of a
non-zero intercept ( ) is that the tangency portfolio changes if the test asset is added to the
17
investment opportunity set and the test can be given two interpretations: if the market index is assumed to be the correct benchmark, it will be an approach to see whether the left-hand side of Eq. (12) is correctly priced in terms of evaluating mutual funds (see, e.g., Cochrane, 2005; Campbell, Lo, and MacKinlay, 1997); on the other hand, if the excess return on any asset is assumed to be correctly priced, the test concerns the mean-variance efficiency of the market portfolio or the proxy of the market portfolio (see, e.g. Campbell, Lo, and MacKinlay, 1997 and Roll’s critique).
For ML estimation, the setting of the starting values is notable in a nonlinear optimization. This is because the algorithms used for optimization are mostly iterative and they need to start somewhere. It is known that the setting of the starting values can influence the success of the using of the algorithms in the sense of arithmetic overflow and dithering. In addition, the algorithms improve the guess of starting values by making small steps, although the default value of the iteration is 3000 times in MATLAB R2013a. Hence, only the local maxima can be found. Appendix C shows the starting values setting principles. The results presented in Table 2 and Table 3 are based on these principles.
Table 2
Estimation results from the APD-CAPM for daily data over 5 years
This table presents the parameter estimates of the APD-CAPM and the empirical 95% confidence intervals of these estimates. CI denotes the empirical 95% confidence interval of the estimate in parentheses. The stock with* means not statistically significant. The results are approximate to 4 decimal places.
̂ CI( ̂ ) ̂ CI( ̂ ) ̂ CI( ̂) ̂ CI( ̂)
18
Table 2(CONTINUED)
̂ CI( ̂ ) ̂ CI( ̂ ) ̂ CI( ̂) ̂ CI( ̂)
INTESA SANPAOLO -0.0640 -0.2203 0.0922 1.5819 1.4330 1.7308 0.4745 0.4230 0.5261 1.1997 1.0884 1.3109 L'OREAL 0.0365 -0.1242 0.1972 0.6786 0.3955 0.9617 0.4882 0.3873 0.5892 1.1114 0.8880 1.3348 LVMH 0.0524 0.0314 0.0734 0.9641 0.9488 0.9794 0.5162 0.5078 0.5245 1.1076 1.0038 1.2114 MUENCHENER RUCK.(XET) 0.0185 -0.0311 0.0680 0.7614 0.7088 0.8140 0.4958 0.4760 0.5157 0.9682 0.4089 1.5276 ORANGE -0.0545 -0.1045 -0.0045 0.7119 0.6865 0.7373 0.4762 0.4675 0.4848 1.0016 0.9116 1.0916 PHILIPS ELTN.KONINKLIJKE 0.0330 0.0216 0.0444 0.9913 0.9848 0.9979 0.4760 0.4657 0.4862 1.0801 0.5911 1.5692 REPSOL YPF* 0.0021 -0.0435 0.0478 1.0164 1.0020 1.0307 0.4994 0.4567 0.5420 0.8737 -6.5483 8.2956 RWE(XET) -0.0708 -0.0762 -0.0653 0.7529 0.7509 0.7548 0.5279 0.5275 0.5283 0.8986 0.8005 0.9967 SAINT GOBAIN -0.0123 -0.0132 -0.0115 1.3649 1.3644 1.3655 0.5031 0.5027 0.5036 0.9351 0.8584 1.0118 SANOFI 0.0302 0.0061 0.0544 0.6879 0.6652 0.7106 0.4987 0.4719 0.5255 1.0634 1.0326 1.0942 SAP(XET) 0.0513 -0.0146 0.1172 0.5559 0.5189 0.5930 0.5007 0.4740 0.5275 1.0657 0.9763 1.1552 SCHNEIDER ELECTRIC 0.0380 -0.0339 0.1100 1.2523 1.1983 1.3064 0.4897 0.4639 0.5154 1.2317 1.1543 1.3090 SIEMENS(XET) 0.0325 -0.2447 0.3097 0.9039 0.6808 1.1271 0.4884 0.3712 0.6056 0.9361 0.7603 1.1118 SOCIETE GENERALE -0.0346 -0.1696 0.1004 1.6896 1.6291 1.7501 0.5063 0.5010 0.5115 0.9934 0.8937 1.0930 TELEFONICA -0.0322 -0.0598 -0.0047 0.8422 0.6332 1.0512 0.5071 0.4858 0.5285 0.9801 0.8925 1.0676 TOTAL 0.0039 -0.0355 0.0433 0.8194 0.7767 0.8620 0.4931 0.4761 0.5100 1.1912 1.0384 1.3439 UNIBAIL-RODAMCO 0.0447 -0.0199 0.1094 0.7768 0.7595 0.7941 0.4921 0.4708 0.5134 1.0879 1.0019 1.1739 UNICREDIT -0.0927 -0.1592 -0.0263 1.5595 1.5564 1.5626 0.5043 0.4891 0.5195 0.9782 0.8848 1.0716 UNILEVER CERTS. 0.0365 0.0320 0.0411 0.4410 0.4407 0.4412 0.4892 0.4877 0.4907 1.0143 0.9326 1.0961 VINCI 0.0176 -0.0452 0.0804 1.0960 1.0804 1.1116 0.4826 0.4383 0.5270 1.1823 1.0023 1.3622 VIVENDI -0.0154 -0.0509 0.0202 0.8322 0.7908 0.8736 0.4917 0.4817 0.5018 1.0704 0.9863 1.1544 VOLKSWAGEN PREF. (XET) 0.1029 -0.2334 0.4392 0.9927 0.9780 1.0074 0.5155 0.4593 0.5717 1.0048 0.9100 1.0996
Note: Due to space limit, the author does not include the estimate of and the corresponding results are available from the author upon request.
In terms of the daily data, Table 2 shows that the results of 49 stocks are not statistically significant at the 5% significant level, whereas Table 3 shows that the results of 43 stocks are not statistically significant at the 5% significant level. As to the results of weekly returns and monthly returns, due to space limit, they are presented in Appendix D.
Table 3
Estimation results from the AEPD-CAPM for daily data over 5 years
This table presents the parameter estimates of the AEPD-CAPM and the empirical 95% confidence intervals of these estimates. CI denotes the empirical 95% confidence interval of the estimate in parentheses. The stock with* means not statistically significant. The results are approximate to 4 decimal places.
̂ CI( ̂ ) ̂ CI( ̂ ) ̂ CI( ̂) ̂ CI( ̂ ) ̂ CI( ̂ )
19
Table 3(CONTINUED)
̂ CI( ̂ ) ̂ CI( ̂ ) ̂ CI( ̂) ̂ CI( ̂ ) ̂ CI( ̂ )
DEUTSCHE BANK(XET) -0.0158 -0.0182 -0.0134 1.3668 1.3654 1.3682 0.5245 0.5243 0.5248 0.9414 0.9407 0.9421 0.8192 0.8190 0.8194 DEUTSCHE POST(XET) 0.0487 0.0369 0.0605 0.8480 0.8003 0.8956 0.4843 0.4774 0.4911 1.0682 0.9359 1.2005 1.0773 0.9476 1.2071 DEUTSCHE TELEKOM(XET) 0.0172 -0.3314 0.3657 0.6226 0.5813 0.6639 0.4953 0.4251 0.5655 0.9646 0.8618 1.0674 0.9522 0.5742 1.3302 E ON(XET) -0.0642 -0.3587 0.2302 0.8659 0.3864 1.3454 0.5889 0.3529 0.8248 1.0617 0.8901 1.2333 0.8709 0.7410 1.0007 ENEL* -0.0213 -2.9749 2.9323 0.9314 -15.8642 17.7271 0.4725 -7.3323 8.2773 0.9544 -1.1887 3.0975 1.0807 -0.0538 2.2152 ENI -0.0027 -0.2602 0.2548 0.8779 0.7139 1.0419 0.5031 0.4662 0.5399 1.0239 0.3421 1.7057 1.0053 0.3188 1.6918 ESSILOR INTL. 0.0529 -0.0083 0.1141 0.3959 0.3523 0.4395 0.4800 0.4552 0.5049 1.1281 1.0430 1.2132 1.1555 1.0183 1.2927 GDF SUEZ -0.0616 -0.8820 0.7589 0.9319 0.0630 1.8007 0.5191 0.5119 0.5263 0.9565 0.8946 1.0184 0.9553 0.8953 1.0152 IBERDROLA -0.0351 -0.0948 0.0246 0.9278 0.9157 0.9398 0.5313 0.5058 0.5568 0.9897 0.8703 1.1091 0.8587 0.7358 0.9817 INDITEX 0.0847 -0.0616 0.2309 0.6859 0.6291 0.7428 0.4769 0.4398 0.5140 1.1603 1.1064 1.2142 1.1802 1.1578 1.2025 ING GROEP* -0.0131 -3.3511 3.3250 1.7333 -15.0424 18.5091 0.5124 -2.5601 3.5850 0.8747 -11.2517 13.0011 0.8228 -4.4043 6.0499 INTESA SANPAOLO -0.0613 -0.2319 0.1092 1.5783 1.5135 1.6431 0.4016 0.2734 0.5297 1.0440 0.9538 1.1341 1.3513 1.3233 1.3792 L'OREAL 0.0378 -0.0552 0.1308 0.6786 0.6458 0.7114 0.4809 0.3670 0.5948 1.0830 0.9262 1.2399 1.1424 1.0292 1.2555 LVMH 0.0518 0.0244 0.0792 0.9712 0.9064 1.0361 0.4696 0.4645 0.4746 1.0154 0.7631 1.2677 1.1621 1.0318 1.2924 MUENCHENER RUCK.(XET) 0.0180 -0.0457 0.0818 0.7699 0.7309 0.8088 0.4891 0.4816 0.4965 0.9517 0.8499 1.0535 0.9867 0.8893 1.0841 ORANGE -0.0548 -0.4920 0.3825 0.7131 0.3932 1.0330 0.3796 0.2998 0.4594 0.8715 0.6737 1.0693 1.1493 0.9576 1.3410 PHILIPS ELTN.KONINKLIJKE* 0.0333 -0.0774 0.1440 0.9908 -2.4563 4.4379 0.4982 0.3492 0.6473 1.1079 -5.8457 8.0615 1.0415 -1.4610 3.5440 REPSOL YPF* -0.0001 -6.3783 6.3781 1.0249 0.5834 1.4664 0.4946 -0.3448 1.3340 0.8570 -11.3505 13.0646 0.8889 -5.8541 7.6320 RWE(XET) -0.0699 -0.8090 0.6692 0.7542 0.0194 1.4890 0.5461 0.3422 0.7499 0.9256 0.7558 1.0953 0.8667 0.6193 1.1140 SAINT GOBAIN -0.0128 -0.1081 0.0825 1.3642 0.9602 1.7683 0.4854 0.4588 0.5119 0.9128 0.7950 1.0307 0.9755 0.8313 1.1198 SANOFI 0.0320 -0.5576 0.6216 0.6925 -3.8792 5.2643 0.4698 0.4644 0.4753 0.9857 -0.2024 2.1739 1.1502 0.7363 1.5641 SAP(XET)* 0.0542 -0.9970 1.1054 0.5493 -0.2813 1.3799 0.4425 -0.5772 1.4623 0.9425 -0.5701 2.4552 1.2392 -16.3975 18.8759 SCHNEIDER ELECTRIC 0.0369 -0.0115 0.0854 1.2519 1.1907 1.3132 0.4922 0.4704 0.5140 1.2440 0.4930 1.9950 1.2204 0.9389 1.5020 SIEMENS(XET) 0.0318 -0.0724 0.1359 0.9094 0.4449 1.3739 0.4706 0.4027 0.5385 0.9092 0.7317 1.0867 0.9791 0.6921 1.2660 SOCIETE GENERALE -0.0343 -0.1364 0.0677 1.6924 1.6422 1.7426 0.5377 0.5158 0.5596 1.0902 0.9662 1.2142 0.9302 0.8298 1.0305 TELEFONICA -0.0319 -0.0882 0.0244 0.8380 -26.8549 28.5309 0.5213 -1.0826 2.1252 1.0083 0.2125 1.8041 0.9408 -2.6411 4.5228 TOTAL 0.0026 -2.9039 2.9091 0.8189 0.0658 1.5719 0.5052 0.0545 0.9560 1.2301 -0.8313 3.2915 1.1512 0.3331 1.9692 UNIBAIL-RODAMCO 0.0449 -0.0668 0.1566 0.7767 0.7415 0.8119 0.4913 0.4650 0.5175 1.0845 0.7523 1.4167 1.0914 0.8710 1.3117 UNICREDIT -0.0919 -0.8628 0.6789 1.5540 0.4720 2.6360 0.5267 0.5171 0.5364 1.0078 0.7668 1.2488 0.9203 0.5612 1.2794 UNILEVER CERTS. 0.0367 -0.0479 0.1213 0.4397 0.4395 0.4400 0.4807 0.4686 0.4928 0.9851 0.8904 1.0797 1.0365 0.7692 1.3038 VINCI 0.0153 -0.1862 0.2168 1.0928 1.0229 1.1628 0.5059 0.4453 0.5665 1.2654 1.1505 1.3803 1.1253 1.0065 1.2442 VIVENDI* -0.0163 -0.1288 0.0963 0.8343 -0.3009 1.9695 0.5166 0.3068 0.7265 1.1163 -2.2789 4.5116 1.0065 -0.3413 2.3543
VOLKSWAGEN PREF. (XET) 0.1030 0.1025 0.1035 0.9898 0.9855 0.9941 0.4613 0.4608 0.4619 0.9106 0.7922 1.0291 1.0721 0.8712 1.2729
Note: Due to space limit, the author does not include the estimate of and the corresponding results are available from the author upon request.
20
̂
. Therefore, the AEPD can capture the asymmetry of tails of returns and thus support the theory discussed in Section 2.3.2, as well as the AIC values and the density plots do.From Table 2, Table 3, and tables in Appendix D, it can be seen that the tail parameter estimates distribute around 1 and that the skewness parameter estimates distribute around 0.5, implying that the returns may follow the Laplace distribution. Fortunately, the Laplace distribution can be tested. If 0.5 lies outside the empirical CI of ̂ or 1 lies outside the empirical CI of ̂, or 1 lies outside the CI of ̂ or ̂ , say the returns statistically significantly do not follow the Laplace distribution at the 5% significance level. Table 2 shows that 26 stock returns significantly do not follow the Laplace distribution, while Table 3 shows that 28 stock returns significantly do not. There are 31 weekly return series and 24 monthly return series that do not follow the Laplace distribution when modeling the return series using the AEPD (see Table D.2 and Table D.4,), whereas only 11 weekly return series and 15 monthly return series do not using the APD. Therefore, more results estimated by the AEPD-CAPM are significant with respect to the Laplace distribution test.
The zero
̂
or non-zerô
estimated by the CAPM implies whether the market portfolio is mean/variance efficient or not, or the market portfolio is the tangency portfolio or not. However, this test is not the main goal of this paper and thus the results estimated by the CAPM (with the IID and jointly normal distributional assumption) are presented in Appendix E. Take the daily returns for example, 28 empirical CIs of̂
estimated by the CAPM
include 0 (see Panel A in Table E),however, 32 empirical CIs of
̂
estimated by the APD-CAPM
include 0(see Table 2) and 43empirical CIs of
̂
estimated by the AEPD-CAPM
include 0(see Table 3). Therefore, the issue whether the market portfolio is mean/variance efficient or whether the market portfolio is the tangency portfolio depends on distributional assumption.As to the most widely used parameter, beta, different models yield different beta estimates for the same stock. Using the ̂ s listed in Panel A of Table E (see Appendix E) as the null hypotheses, in terms of the daily returns, 19 corresponding ̂ s estimated by the APD-CAPM are significant (see the column of CI
(
̂
) in Table 2) while 11 corresponding ̂ s estimated by the AEPD-CAPM aresignificant (see the column of CI
(
̂
) in Table 3). An application of beta estimates will be discussed in Section 5.4.3. Goodness of fit
21
approximate the reality and thus the correct model does not exist in these three alternative models; therefore, the AIC is the most applicable. In addition, the Kernel Density Estimation is used to compare the density plots.
4.3.1. The Akaike Information Criterion
Based on Akaike (1974), the AIC is defined as
( ̂) (23) where
̂
is the value of the log-likelihood function, evaluated at and k is thenumber of components of ̂ . The smaller the AIC value the better the model is.
Due to space limit, the AIC values are presented in Appendix F. For daily returns (see Panel A in Table F), it can be seen that the asymmetric CAPMs are better than the CAPM, because every AIC value of the CAPM is the largest. Moreover, the APD-CAPM performs best with 35 smallest AIC values. As to the weekly returns, see Panel B, compared with the CAPM, 48 out of 50 AIC values (without BAYER (XET) and TOTAL) obtained by the APD-CAPM are smaller and 47 out of 50 AIC values (without BAYER (XET), and SIEMENS (XET) and TOTAL) obtained by the AEPD-CAPM are smaller. In addition, 28 AIC values obtained by the APD-CAPM are the smallest while 20 obtained by the AEPD-CAPM are the smallest. From Panel C, for monthly returns, 29 AIC values obtained by the CAPM are the smallest, 10 AIC values obtained by the AEPD-CAPM are the smallest, and 11 AIC values obtained by the APD-CAPM are the smallest. Therefore, the CAPM is the best model for monthly returns while the APD-CAPM is the best for daily and weekly returns.
4.3.2. The Kernel Density Estimation
22
Fig. 2 The PDF plots of actual residuals and the given distribution (ALLIANZ)
The black solid line represents the empirical distribution while the red dotted line represents the given distribution. The sample period is from 2/1/2009 to 3/1/2014. See the plots of ALLIANZ, compared with the normal distribution, the power distributions demonstrate a very close fit, although the values of tails parameters in the APD-CAPM and the AEPD-CAPM are similar ( ̂ 0.8851 and ̂ 0.8973, ̂ 0.8824, see Table 2 and Table 3 in Section 4.2).Hence, the power distributions can better account for the asymmetry and fat-tails.
Hence, both the AIC values and the density plots show that the asymmetric and fat-tailed CAPMs (the APD-CAPM and the AEPD-CAPM) perform better than the CAPM when modeling the daily and weekly returns.
5. Minimum Variance Portfolios and the recursive analysis
In this section, the beta estimates yielded by different distribution types of CAPMs are applied to MVP construction and the recursive analysis offers a backtest for simulating the portfolio strategy on monthly basis as it would be done in practice.
5.1. The minimum variance portfolio
Rational investors are risk averse and thus they are willing to hold a diversified portfolio instead of a certain security to reduce risk. According to Plantinga (2013a), almost all securities have a correlation with each other below one, so it is possible to create portfolios that have lower risk at a given level of expected return than a single security with the same level of expected return. According to Thomas and Gup (2010), the standard deviation of a single security does not implies its contribution to a portfolio’s risk. Thus, another measure of risk (i.e. the systematic risk) provided by the CAPM can be used to construct portfolios and the non-systematic risk (i.e. the unique risk) can be eliminated by diversification.
23
that low volatility stocks come with higher returns(e.g., Ang, Hodrick, Xing, and Zhang ,2006; Bali and Cakici,2008). Hence, many investors who wish to invest in low-risk or ‘defensive’ stocks believe that the MVPs concentrated in low volatility stocks will perform well, especially in times of downward market. Moreover, the only required inputs in the optimization of the MVPs are correlations and volatilities only depending on risk parameters (see e.g., Amenc and Martellini, 2002). It is well known that estimating the expected returns is difficult; however, the composition of the MVP is not affected by expected returns (see, e.g., Plantinga, 2013a).
The asset allocation problem involves the maximization of a quadratic function of the decision variables, subject to a set of linear constraints, some of which are inequalities (see, e.g., Markowitz, 1956; Sharpe, 1963; Sharpe, 1970; Sharpe, 1971; Elton et al, 1978). However, according to Sharpe (1978), the approaches developed by Sharpe (1963; 1970; 1971) and Elton et al(1978) for a simplified model hardly accommodate a different model of covariance relationships. Therefore, the algorithm developed by Sharpe (1978) is employed to construct MVP, exploiting the simplicity of the constraints while accepting as much complexity as desired in the covariance matrix.
However, according to Sharpe (2007b), in terms of the MVP, an investor who wishes to minimize risk will have a risk tolerance of zero, no matter how much expected return is sacrificed in the process. Therefore, the asset allocation problem of the MVP in terms of relative risk aversion can be written as follows:
[ ] (24)
subject to:
where denotes relative risk aversion, is a vector of decision variables i.e. the asset holdings, expressed as proportionate values, to represent a portfolio, denotes a vector of asset expected returns, denotes a covariance matrix, denotes a vector of lower bounds, and denotes a vector of upper bounds. Only non-negative values of are allowed if short sales are not allowed, hence, and (The Proof of Eq.(24) , see Appendix A).
Sharpe offers the MATLAB gqp function15 to obtain the solution to a similar problem on his website. Based on his work, the author uses function, [ ] 16
to conduct the recursive analysis on the MVPs, where denotes the minimum variance mix vector, denotes the expected return of the portfolio, denotes the variance of the portfolio return, denotes the vector of the expected return which can be calculated by the mean of historical returns, and denotes the initial feasible mix vector. denotes the covariance matrix.
The covariance matrix is generated by employing the estimates for systematic risks, the betas
15
The MATLAB code is available from Sharpe’s official website http://www.stanford.edu/~wfsharpe/mia/opt/mia_opt1.htm.
http://www.stanford.edu/~wfsharpe/mat/gqp.txt. Last access date: 05-06-2014.
16
24
yielded by each type of CAPM. These estimates are essential input for the covariance matrix17 estimator of the returns for the single-index model (SIM), because the portfolio selection process is simplified by assuming only one source of systematic risk, i.e. the Market Risk. Moreover, according to Plantinga (2013b), the covariance estimate of the SIM’s descriptive ability is better than the one generated by the average correlation model, and its predictive ability is better than the historical covariance matrix and the one yielded by the multi-index model. Besides, the full covariance matrix estimated without restrictions is unstable, and imposing constraints is an eminently sound statistical idea18
.
Hence, the element in the SIM covariance matrix estimator can be defined as follows:
{
(25)
where denotes the variance of security i’s return, denotes the variance of the market index return. Note and are those yielded by the capital asset pricing models discussed in this paper and are not yielded by the SIM discussed in Elton et al(2011).
5.2. Recursive analysis over five years
From a pragmatic point of view, recursive analysis via computer programming is conducted to know how well the MVPs perform using the SIM covariance relationship. To simulate the real asset allocation procedures as much as possible, MVPs with non-negative weights are determined recursively for monthly data, using all three return distributions. All estimates and weights are recalculated when new cross-sectional monthly returns are added to the data set. This yields five-year monthly MVP return series for each distributional approach (see Table G in Appendix G, it shows the MVP returns from 14/6/2009 to 14/4/2014.).
The data from 14/7/2001 to 14/4/2014 are disposed in line with data description section (see Section 4.1.3). However, due to data availability, GDF SUEZ, which is only available from 29/7/2005, is excluded. After experimenting, 95 period’s data can ensure the specification of models estimated using the Method of ML. Hence, the first recursion starts from 14/7/2001 until 14/5/2009 to generate the minimum variance mix vector for period 14/6/2009, and then the second recursion starts from 14/7/2001 to 14/6/2009 to re-calculate the MVP for period 14/7/2009. Repeat the recursion until all returns are used to construct the MVP for period . Table 4 shows the descriptive statistics of these portfolio return distributions yielded by different approach to beta estimates.
17
The author thanks Prof.Auke Plantinga for suggesting the single-index correlation structure. 18
25 1 1 8 9 13 14 11 2
MVP return distribution( the normality approach) Frequency 1 3 7 9 12 14 9 4
MVP return distribution(the APD approach） Frequency 1 3 5 11 13 15 9 2
MVP return distribution(the AEPD approach）
Frequency
Table 4
The descriptive statistics of portfolio return distributions
This table presents the descriptive statistics of portfolio return distributions. From this table, the median and the mean of the portfolio return series constructed by the AEPD approach to beta estimates are larger than those constructed by the APD approach the normality approach, whereas its standard deviation is the smallest. The data are log returns that have the advantage of time-additivity. The sum of the portfolio returns series via the AEPD approach is higher than the other alternatives. The results are approximate to 4 decimal places,*means the actual value is smaller than the approximate value presented in table.
MVPs—the normality approach MVPs—the APD approach MVPs—the AEPD approach
Median 0.0152 0.0149 0.0158 Maximum 0.0890 0.0854 0.0879 Minimum -0.0738 -0.0694 -0.0712 Mean 0.0116 0.0116 0.0118 Standard Deviation 0.0364 0.0364* 0.0360* Skewness -0.2093 -0.1698 -0.1914 Kurtosis 2.4934 2.4036 2.4243 Sum 0.6824 0.6857 0.6980 Sum Sq. Dev. 0.0769 0.0769 0.0750 Observations 59 59 59
The conventional statistics for the MVPs, the Sharpe Ratio (the ratio of expected return minus the riskless rate to standard deviation), are presented in Appendix H. Fig. 3 shows the MVP return distributions based on different distributional assumptions.
Fig. 3 Portfolios return distributions.
26 6. Conclusion and discussion
According to Thomas and Gup (2010) , the non-normality of returns has reemerged due to the recent 07-08 financial crisis, and two questions arise if applications of the normal distribution in finance are questionable: (1) What is an accurate assumption regarding the distribution of security returns? and (2) What does the assumption imply for practitioners, especially those in the field of security valuation and portfolio management? To answer these questions, two new asset-pricing models are proposed and their application in minimum variance portfolio is discussed in this paper.
Considering asymmetry, heavy tails, leptokurtic or fat-tails of returns, two recent asymmetric distributions, the APD proposed by Komunjer(2007) and the AEPD proposed by Zhu and Zinde-Walsh(2009) are used for modeling returns. This is because the APD and the AEPD are capable of accommodating asymmetry and fat-tails. The APD extends the GPD in the sense of capturing the asymmetry of financial data. The AEPD extends the SEPD in the sense of capturing the asymmetry in tails to offer a more flexible tail shape due to two tail parameters (see Fig.2 in Section 4.3.2).
In this paper, three different distribution types of CAPMs analyze the daily, weekly, and monthly returns of the EURO STOXX 50 index and its constituents from 2/1/2009 to 3/1/2014. The statistical evidence show that the APD-CAPM and the AEPD-CAPM are better fitted daily and weekly total returns of the Eurozone stock market, especially the APD-CAPM. According to the empirical 95% confidence intervals of estimates, for higher frequency financial data, the majority of the results of the asymmetric CAPMs are statistically significant due to the normality test and thus the IID normal distribution hypothesis can be rejected in most cases. Besides, the Laplace distribution hypothesis test shows that more results from AEPD-CAPM are significant. Moreover, according to Zeckhauser and Thompson (1970), it is desirable to estimate the parameters that describe the power distribution if the accuracy of estimation is important. It is obvious that the coefficients estimates estimated by the CAPM, the APD-CAPM, and the AEPD-CAPM are so different that, as Zeckhauser and Thompson (1970) state, the effect caused by estimating both the coefficients and the parameters describing the power distribution on regression coefficients cannot be small. Specifically, most of the alpha estimates estimated by the CAPM are significant in terms of the null hypothesis of
̂
while lesŝ
s aresignificant by either the APD-CAPM or the AEPD-CAPM. Hence, the results of alpha test depend on the distributional assumptions. In addition, when the null hypothesis is that the beta estimate from the AEPD-CAPM or from the APD-CAPM equals the beta estimate from the CAPM, more beta estimates from the APD-CAPM or the AEPD-CAPM are significant regarding to higher frequency returns. That is, the smaller the time intervals, the more significant results the asymmetric CAPMs have.
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recursive beta estimates are used for the estimated covariance matrix of the returns for the SIM. Using all three return distributions and re-calculating all estimates and weights anew, five-year monthly MVP returns for each distributional approach show that the results are quite comparable. The approaches to beta estimates generated by the asymmetric CAPMs can reduce the volatility of the constructed MVP monthly return series. Moreover, the sums of the log returns (including dividends) based on the asymmetric return distributions are higher than the sum based on the symmetric return distribution.
Since the maximum likelihood can performs well when the sample size is infinite according to asymptotic theory (see Mooney and Duval, 1993). From the sample size point of view, higher frequency returns can be preferable within the same time frame. This leaves us to an open question: what effects do the frequency, daily, weekly and monthly have on the minimum variance portfolio returns19?
For finite samples, like the monthly returns on minimum variance portfolios, the bootstrap can be used to assess the statistical stability of the outcomes based on the recursive analysis20. By resampling the original data based on the recursive analysis, the bootstrap statistic can be obtained. As can be seen from Table 4 and Fig.3, the monthly MVP returns are skewed. Thus, the Bias-corrected and accelerated ( ) method developed by DiCiccio and Efron(1996) can be used to give for the difference between the average returns of the minimum variance portfolios. This open question can be answered by further research.
Due to data accessibility and the time strain, the asymmetric CAPMs proposed in this paper are only applied into the Eurozone stock market. However, these models can be used for a broader data set and applied to other asset classes, such as bonds, other fixed income securities, and real estate. In addition, since these asymmetric and fat-tailed distributions discussed in this paper can be applied into the fields of corporate valuation and portfolio management, the relevant theory can be developed, as Rachev and Mittnik (2000)’s work for the Stable Paretian, which is based on the studies of Fama (1971), Ross (1991) and other academics.
Finally and most importantly, other state-of-the-art studies of distributions, such as a generalized asymmetric Student-t distribution (see Zhu and Galbraith, 2010), can be applied using the same model-building process. Therefore, to investigate the most appropriate return distribution in terms of different distribution types of CAPMs, the further research could be a comparison study of the asymmetric distributions, such as the APD, the AEPD, and a generalized asymmetric Student-t distribution, with the symmetric Stable Paretian using a broader market index, like the S&P 500 or the CRSP market index.
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Due to curiosity, the author once ran the weekly and daily frequency returns, however, it was limited by the computation speed of computers,15 computers cannot finish it in 14 hours. Further research can be done to investigate the frequency effects.
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